Algebra 2B — Exponentials, Logs & Sequences

Exponential growth & decay

ELI5: exponential growth means multiplying by the same factor every step (not adding). At 7 % a year, money is ×1.07 each year — so it accelerates: the more you have, the faster it grows.

P_{0} = 1000

starting amount

rate = 0.07

growth per period — 7 %

P_0 * (1 + rate)^x

the growth curve — every doubling takes the SAME number of periods

Real-world hook: compound interest, viral spread, and — with a factor below 1 — radioactive decay, cooling coffee, and a drug clearing your bloodstream.

Try it yourself: how many years for $1000 at 5 % to reach $2000? (Solve 1000·1.05ᵗ = 2000.)

t_{you} = \frac{\mathrm{log}\left(2\right)}{\mathrm{log}\left(1.05\right)}

14.2067

✏️ Your turn: find the doubling time t. The check grows \$1000 at 5 % for your t and sees if it reaches \$2000. (Hint: t = log 2 / log 1.05.)

✓ pass

\mathrm{abs}\left(1000 \cdot 1.05^{t_{you}} - 2000\right) < 1

green when the balance really doubles